- #1
Sentin3l
- 118
- 6
Homework Statement
Given a uniform sphere of mass M and radius R. Use integral calculus and start with a mass dm in the sphere. Calculate the work done to bring in the remainder of the mass from infinity. By this technique show that the self-potential energy of the mass is:
P = -\frac{3}{5} \frac{GM^{2}}{R}
Homework Equations
W = \int\vec{F} \bullet d\vec{r}
F = \frac{GMm}{r^{2}}
F = \frac{GMm}{r^{2}}
The Attempt at a Solution
First let me say that this is a cosmology question. I began by considering a differential mass near or at the center of the sphere. Using the above equations for force and work, I derived:
W = - \int \frac{GM(dm)}{r^{2}} \hat{r}
Since the sphere is uniform, it has a constant mass to radius ratio λ = \frac{M}{R} = \frac{dm}{dr}. So using this I found:
W = -λ\int \frac{GM}{r^{2}}dr = -3λ \frac{GM}{r^{3}}
If we substitute λ = \frac {M}{R} and r=R, we get the result:
W = -3 \frac{GM^{2}}{R^{3}}
Here is where i think I went wrong, I don't know if I need to deal with \hat{r} and if so, I'm not sure how to approach that.
I think that once I get the work, you use the work-energy theorem, and intial/final KE is 0 so the potential energy equals the work, please correct me if I'm wrong in that.
